How to Find Gravitational Acceleration without Mass: Several Approaches and Problem Examples


The article discusses about several approaches on how to find gravitational acceleration without mass. 

The gravitational acceleration g does not depend on the small mass. Hence it is neglected while calculating g. The formulas and concepts of an inclined plane, buyout force, Kepler’s law, and spherical symmetric bodies help us to find the gravitational acceleration without small mass. 

While calculating the value of gravitational acceleration g in the previous article, we learned that the g value depends only on the high mass M of gravitating bodies like earth or other planets, which have a constant mass value for all other objects they attract. In contrast, the mass of the object that attracts towards gravitating body is overlooked

Let’s find the value of g by several approaches and problems, employing different laws where small mass is overlooked. 

How to find Gravitational Acceleration
How to calculate Gravitational Acceleration

Read more about How to find Mass from Gravitational Acceleration.

How to find Gravitational Acceleration without Mass using Buyout Force

Let’s see how buyout force from fluid helps us to find gravitational acceleration without mass.

The fluid exerts the buyout force on an object to oppose its weight ‘mg’ whether it sinks or floats. When the buyout force greater than weight or gravity force, an object floats on fluid. If the gravity force larger, an object sinks in the fluid. 

How to find Gravitational Acceleration without Mass?
How to find Gravitational Acceleration
without Mass
How to find gravitational acceleration using Buyout force
Buyout Force and Gravity Force
(credit: shutterstock)

When we dive into the swimming pool, the water exerts an upward force on our bodies. The same upward we experience when we swim under the water. The upward force is the pressure by water exerted is called ‘buyout force’.  

As we go deeper into the water, it exerts high pressure on our bottom side and low pressure on our top side.  

How to find gravitational acceleration using Buyout force
Pressure on Top and Bottom Sides

To calculate the total buyout force acting on an object, first, calculate buyout force on the top and bottom sides individually and then subtract them.

Fb = Fbottom – Ftop   ……………… (*)

The exerted pressure is the applied force per unit area on which the applied force is distributed.

P=F/A

F=PA

Depending on its weight, an object will float or sink in the water. That’s why buyout force is related to the gravity force Fg

Fg= mg

Substituting above gravity force into equation (1),

P=mg/A

Whereas mass = volume x density i.e.,

Where, height h = V/A

The above formula is the formula of Hydrostatic Gauge Pressure exerted by water due to gravity force.

How to find gravitational acceleration using Buyout force
Hydrostatic Gauge Pressure Formula

As per equation (1), the buyout force on top sides is

Ftop = PtopA

Substituting gauge pressure value (2),

Similarly, the buyout force on bottom side is 

Equation (*) becomes,

Where (hbottom – htop) is the exact height ‘h’ of an object.

Therefore,

Where ‘Ah’ is the volume of the displaced water.

Hence,

That’s how using buyout force; we can calculate the gravitational acceleration g in terms of density and volume.

Read more about Types of Forces.

When we placed a cube into the tub filled with water, it displaced the water to 1.55 liters. The buyout force experienced on a cube is 15 N by the water, which has a density of about 1000 kg/m3. Calculate the gravitational acceleration of the cube inside the water.

Given:

ρ = 1000 kg/m3

V = 1.53 litres

Since 1 m3 = 1000 litres, so 1.53 litres = 0.00153 m3

Fb = 15 N

To Find: g=?

Formula:

Solution:

The gravitational acceleration on a cube is calculated using the buyout force formula as,

Substituting all values,

15=1000*g*0.00153

g=15/1.53

g = 9.803

The gravitational acceleration on a cube is 9.80 m/s2.

How to find Gravitational Acceleration without Mass using Spherically Symmetric Body

Let’s see how the gravitational acceleration is calculated without mass using spherically symmetric body.

In a spherically symmetric body, all of its mass is concentrated at one point. By substituting the large Mass M value of the spherically symmetric body into the gravity force, we can obtain the gravitational acceleration in terms of its density ρ0.

How to find Gravitational Acceleration using Spherically Symmetric Bodies
Gravitational Acceleration g Formula
for Spherically Symmetric Body

We have obtained the gravitational acceleration without mass as by substituting the mass value first into gravity force by the law of gravitation, and then into gravity force by Newton’s second law

To know how we derived the above formula, read the article here.

Calculate the gravitational acceleration of the astronaut walking on the moon. The moon has a density of about 104 kg/cm3, and the distance between the center of mass of both astronaut and moon is 1.74 x 106m.

Given:

G = 6.67 x 10-11 Nm2/kg2

r = 1.74 x 106m

σ = 104 g/cm3

To Find: g=?

Formula:

g=(4/3)Gσr

Solution:

If we consider the moon as a spherically symmetric body, then

The gravitational acceleration on the astronaut is calculated as,

g=(4/3)Gσr

Substituting all values,

(46.42*10-1)/3

g =1.54

The gravitational acceleration of the astronaut walking on the moon is 1.54 m/s2.

Read more about How to Calculate Mass from Force and Distance.

How to find Gravitational Acceleration without Mass using an Inclined plane

Let’s see how an inclined plane supports us to find gravitational acceleration without mass.

Because of the triangular shape, it requires less force to accelerate any object on an inclined plane. Therefore, the accelerated incline object drops a slightly different value than gravitational acceleration g. If we know the falling object’s acceleration and incline angle, we can find the g value without mass. 

How to find gravitational acceleration  using an Inclined plane
Gravitational Acceleration g Formula
on an Inclined Plane

When an object accelerates horizontally along the earth’s surface, it accelerates equally to the value of g, which is 9.8 m/s2. But if we inclined the horizontal surface at a certain angle, its acceleration value becomes slightly different from the g value.  

Therefore, the gravity force ‘mg’ is resolved into pair components on a frictionless inclined plane. The one component acts perpendicular to the plane ‘mgcosθ’, and the other is parallel to the plane ‘mgsinθ’. The normal force counterpart ⊥ components force by acting opposite to it. 

So the only || component of gravity force accelerates an object on an inclined plane.

As per Newton’s second law, the net force acting on an object, 

Fnet=ma

a=Fnet/m

Since the net force acting on an inclined plane,

F|| = mgsinθ

Therefore, Newton’s second law becomes

a=mgsinθ/m

a=gsinθ

g=a/sinθ

The above equation shows that depending on an incline angle, the falling object’s acceleration value varies from constant g value gravitational acceleration.

Read more about Work Done on an Inclined Plane.

Calculate the gravitational acceleration on the ball accelerated downward at 6 m/s2 on the surface that is inclined at 38°.

Given:

a = 6 m/s2

θ = 38°

To Find: g=?

Formula:

g=a/sinθ

Solution:

The gravitational acceleration on the ball falling on an inclined plane is calculated as,

g=a/sinθ

Substituting all values,

g=6/sin 380

g=6/0.615

g=9.75

g = 9.75

The gravitational acceleration on the falling ball is 9.75 m/s2.

Calculate the acceleration of the box sliding down on an inclined ramp at 87°.

Given:

θ = 87°

g = 9.8 m/s2

To Find: a =?

Formula:

a=gsinθ

Solution:

The acceleration of the box sliding down is calculated as,

a=gsinθ

a=9.8*sin 870

a=9.8*0.998

a=9.78

Substituting all values,

a = 9.78

The acceleration of the box sliding down is 9.78 m/s2.


Manish Naik

Hello, I'm Manish Naik completed my MSc Physics with Solid-State Electronics as a specialization. I have three years of experience in Article Writing on Physics subject. Writing, which aimed to provide accurate information to all readers, from beginners and experts. In my leisure time, I love to spend my time in nature or visiting historical places. I am honoured to be part of LambdaGeeks. Looking forward to connecting you through LinkedIn - https://www.linkedin.com/in/manish-ashok-naik/ Also, for Maharashtra travel guide and heritage conservation articles, visit my website Wandering Maharashtra - https://wanderingmaharashtra.com/travel-blogs/

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